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The stealth mechanics in BioShock are relatively simplistic, and especially at later levels I rarely find it useful. However, the rules are:

Yes, noise matters. Enemies will hear you moving; the slower you move the quiter you are. They'll also hear your weapon discharge. AFAICT the wrench is "silent" but beating people in the head with another weapon is not.

They will also see you, though I have not done extensive testing on the limits of their line of sight. In general, to get close enough to wrench them to death you need to be almost directly behind them, though I've taken splicers out unseen from the side with ranged weapons.

Once an enemy spots you, you are unlikely to get back into hiding. The splicers are very loud and yell at you constantly while they attack you, which will alert other nearby splicers to your presence. (Not all splicers will pay attention, though, so you may be able to start sneaking again once the current wave of splicers is dead.)

There aren't any specific enemies with unique behavior towards stealth, but there are some enemies that are difficult/impossible to sneak up on. In certain areas, other characters will announce over the PA system that you are in the area and send the splicers after you. Boss splicers also seem to frequently know you're coming and/or expect you, so they'll know where you are from the start.

(I have no idea if difficulty matters.)

Once splicers are altered to your possible presence, they may begin to actively seek you out. You can usually tell when a this happens, as their idle animations are different from their seeking animations, and while looking for you they will often taunt you out of hiding.

Certain tonics will help with a stealthy approach:

The two Wrench Lurker tonics you mentioned make it easier to sneak up on enemies to within melee range, and boost your wrench damage. However, you don't need to be stealthy to use these tonics; "unaware" here should probably be more like "distracted", as things like Insect Swarm and Target Dummy will also provide the bonus.

Natural Camouflage can help you get close to an enemy as well. Note that enemies that already know where you are won't be fooled by your camo, so you can't use this to go back into stealth once it's been lost.

Nice answer. Have you played BioShock 2 too? If yes, you might also want to answer my similar question about it.
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galacticninjaJan 27 '13 at 6:52

Also, do you know if light levels (hiding in shadows or dark places) help in avoiding detection?
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galacticninjaJan 27 '13 at 6:54

I have played BS2 but I have never done much stealthy playing so I don't know for sure. I'm pretty sure that the light/dark areas in BS1 do matter but it might just be expectation bias on my part.
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Michael EdenfieldJan 27 '13 at 14:46

Do you know if wearing Big Daddy boots will affect stealth? (Once Jack gets the Big Daddy boots at Point Prometheus, Jack will wear it, and the sound of his footsteps will change to that of a Big Daddy's.)
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galacticninjaFeb 6 '13 at 8:36